<s>
A	O
rotation	O
is	O
different	O
from	O
other	O
types	O
of	O
motions	O
:	O
translations	B-Algorithm
,	O
which	O
have	O
no	O
fixed	O
points	O
,	O
and	O
(	O
hyperplane	O
)	O
reflections	O
,	O
each	O
of	O
them	O
having	O
an	O
entire	O
-dimensional	O
flat	O
of	O
fixed	O
points	O
in	O
a	O
-dimensional	O
space	O
.	O
</s>
<s>
Mathematically	O
,	O
a	O
rotation	O
is	O
a	O
map	B-Algorithm
.	O
</s>
<s>
All	O
rotations	O
about	O
a	O
fixed	O
point	O
form	O
a	O
group	O
under	O
composition	B-Application
called	O
the	O
rotation	O
group	O
(	O
of	O
a	O
particular	O
space	O
)	O
.	O
</s>
<s>
But	O
in	O
mechanics	O
and	O
,	O
more	O
generally	O
,	O
in	O
physics	O
,	O
this	O
concept	O
is	O
frequently	O
understood	O
as	O
a	O
coordinate	O
transformation	O
(	O
importantly	O
,	O
a	O
transformation	O
of	O
an	O
orthonormal	B-Algorithm
basis	O
)	O
,	O
because	O
for	O
any	O
motion	O
of	O
a	O
body	O
there	O
is	O
an	O
inverse	O
transformation	O
which	O
if	O
applied	O
to	O
the	O
frame	O
of	O
reference	O
results	O
in	O
the	O
body	O
being	O
at	O
the	O
same	O
coordinates	O
.	O
</s>
<s>
A	O
representation	O
of	O
rotations	O
is	O
a	O
particular	O
formalism	O
,	O
either	O
algebraic	O
or	O
geometric	O
,	O
used	O
to	O
parametrize	O
a	O
rotation	O
map	B-Algorithm
.	O
</s>
<s>
The	O
"	O
improper	B-Algorithm
rotation	I-Algorithm
"	O
term	O
refers	O
to	O
isometries	O
that	O
reverse	O
(	O
flip	B-Algorithm
)	O
the	O
orientation	O
.	O
</s>
<s>
In	O
the	O
language	O
of	O
group	O
theory	O
the	O
distinction	O
is	O
expressed	O
as	O
direct	O
vs	O
indirect	O
isometries	O
in	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
,	O
where	O
the	O
former	O
comprise	O
the	O
identity	O
component	O
.	O
</s>
<s>
Any	O
direct	O
Euclidean	O
motion	O
can	O
be	O
represented	O
as	O
a	O
composition	B-Application
of	O
a	O
rotation	O
about	O
the	O
fixed	O
point	O
and	O
a	O
translation	B-Algorithm
.	O
</s>
<s>
Composition	B-Application
of	O
rotations	O
sums	O
their	O
angles	O
modulo	O
1	O
turn	O
,	O
which	O
implies	O
that	O
all	O
two-dimensional	O
rotations	O
about	O
the	O
same	O
point	O
commute	O
.	O
</s>
<s>
Any	O
two-dimensional	O
direct	O
motion	O
is	O
either	O
a	O
translation	B-Algorithm
or	O
a	O
rotation	O
;	O
see	O
Euclidean	O
plane	O
isometry	O
for	O
details	O
.	O
</s>
<s>
Any	O
rotation	O
about	O
the	O
origin	O
can	O
be	O
represented	O
as	O
the	O
composition	B-Application
of	O
three	O
rotations	O
defined	O
as	O
the	O
motion	O
obtained	O
by	O
changing	O
one	O
of	O
the	O
Euler	O
angles	O
while	O
leaving	O
the	O
other	O
two	O
constant	O
.	O
</s>
<s>
They	O
constitute	O
a	O
mixed	O
axes	O
of	O
rotation	O
system	O
because	O
angles	O
are	O
measured	O
with	O
respect	O
to	O
a	O
mix	O
of	O
different	O
reference	B-General_Concept
frames	I-General_Concept
,	O
rather	O
than	O
a	O
single	O
frame	O
that	O
is	O
purely	O
external	O
or	O
purely	O
intrinsic	O
.	O
</s>
<s>
When	O
one	O
considers	O
motions	O
of	O
the	O
Euclidean	O
space	O
that	O
preserve	O
the	O
origin	O
,	O
the	O
distinction	O
between	O
points	O
and	O
vectors	O
,	O
important	O
in	O
pure	O
mathematics	O
,	O
can	O
be	O
erased	O
because	O
there	O
is	O
a	O
canonical	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
between	O
points	O
and	O
position	O
vectors	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
vector	O
description	O
of	O
rotations	O
can	O
be	O
understood	O
as	O
a	O
parametrization	O
of	O
geometric	O
rotations	O
up	O
to	O
their	O
composition	B-Application
with	O
translations	B-Algorithm
.	O
</s>
<s>
A	O
motion	O
that	O
preserves	O
the	O
origin	O
is	O
the	O
same	O
as	O
a	O
linear	B-Architecture
operator	I-Architecture
on	O
vectors	O
that	O
preserves	O
the	O
same	O
geometric	O
structure	O
but	O
expressed	O
in	O
terms	O
of	O
vectors	O
.	O
</s>
<s>
In	O
components	O
,	O
such	O
operator	O
is	O
expressed	O
with	O
orthogonal	B-Algorithm
matrix	I-Algorithm
that	O
is	O
multiplied	O
to	O
column	O
vectors	O
.	O
</s>
<s>
Thus	O
,	O
the	O
determinant	O
of	O
a	O
rotation	O
orthogonal	B-Algorithm
matrix	I-Algorithm
must	O
be	O
1	O
.	O
</s>
<s>
The	O
only	O
other	O
possibility	O
for	O
the	O
determinant	O
of	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
is	O
,	O
and	O
this	O
result	O
means	O
the	O
transformation	O
is	O
a	O
hyperplane	B-Algorithm
reflection	I-Algorithm
,	O
a	O
point	B-Algorithm
reflection	I-Algorithm
(	O
for	O
odd	O
)	O
,	O
or	O
another	O
kind	O
of	O
improper	B-Algorithm
rotation	I-Algorithm
.	O
</s>
<s>
Matrices	O
of	O
all	O
proper	B-Algorithm
rotations	I-Algorithm
form	O
the	O
special	O
orthogonal	O
group	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
to	O
carry	O
out	O
a	O
rotation	O
using	O
a	O
matrix	O
,	O
the	O
point	O
to	O
be	O
rotated	O
counterclockwise	O
is	O
written	O
as	O
a	O
column	O
vector	O
,	O
then	O
multiplied	O
by	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
calculated	O
from	O
the	O
angle	O
:	O
</s>
<s>
The	O
matrix	O
is	O
a	O
member	O
of	O
the	O
three-dimensional	O
special	O
orthogonal	O
group	O
,	O
,	O
that	O
is	O
it	O
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
with	O
determinant	O
1	O
.	O
</s>
<s>
That	O
it	O
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
means	O
that	O
its	O
rows	O
are	O
a	O
set	O
of	O
orthogonal	O
unit	O
vectors	O
(	O
so	O
they	O
are	O
an	O
orthonormal	B-Algorithm
basis	O
)	O
as	O
are	O
its	O
columns	O
,	O
making	O
it	O
simple	O
to	O
spot	O
and	O
check	O
if	O
a	O
matrix	O
is	O
a	O
valid	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Above-mentioned	O
Euler	O
angles	O
and	O
axis	O
–	O
angle	O
representations	O
can	O
be	O
easily	O
converted	O
to	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
quaternion	O
can	O
be	O
related	O
to	O
the	O
rotation	O
vector	O
form	O
of	O
the	O
axis	O
angle	O
rotation	O
by	O
the	O
exponential	O
map	B-Algorithm
over	O
the	O
quaternions	O
,	O
</s>
<s>
More	O
generally	O
,	O
coordinate	B-General_Concept
rotations	I-General_Concept
in	O
any	O
dimension	O
are	O
represented	O
by	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
all	O
orthogonal	B-Algorithm
matrices	I-Algorithm
in	O
dimensions	O
which	O
describe	O
proper	B-Algorithm
rotations	I-Algorithm
(	O
determinant	O
=	O
+1	O
)	O
,	O
together	O
with	O
the	O
operation	O
of	O
matrix	O
multiplication	O
,	O
forms	O
the	O
special	O
orthogonal	O
group	O
.	O
</s>
<s>
Matrices	O
are	O
often	O
used	O
for	O
doing	O
transformations	O
,	O
especially	O
when	O
a	O
large	O
number	O
of	O
points	O
are	O
being	O
transformed	O
,	O
as	O
they	O
are	O
a	O
direct	O
representation	O
of	O
the	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
Projective	B-Algorithm
transformations	I-Algorithm
are	O
represented	O
by	O
matrices	O
.	O
</s>
<s>
They	O
are	O
not	O
rotation	B-Algorithm
matrices	I-Algorithm
,	O
but	O
a	O
transformation	O
that	O
represents	O
a	O
Euclidean	O
rotation	O
has	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
in	O
the	O
upper	O
left	O
corner	O
.	O
</s>
<s>
Also	O
in	O
calculations	O
where	O
numerical	B-Algorithm
instability	I-Algorithm
is	O
a	O
concern	O
matrices	O
can	O
be	O
more	O
prone	O
to	O
it	O
,	O
so	O
calculations	O
to	O
restore	O
orthonormality	B-Algorithm
,	O
which	O
are	O
expensive	O
to	O
do	O
for	O
matrices	O
,	O
need	O
to	O
be	O
done	O
more	O
often	O
.	O
</s>
<s>
For	O
odd	O
,	O
most	O
of	O
these	O
motions	O
do	O
not	O
have	O
fixed	O
points	O
on	O
the	O
-sphere	O
and	O
,	O
strictly	O
speaking	O
,	O
are	O
not	O
rotations	O
of	O
the	O
sphere	O
;	O
such	O
motions	O
are	O
sometimes	O
referred	O
to	O
as	O
Clifford	O
translations	B-Algorithm
.	O
</s>
<s>
A	O
generalization	O
of	O
a	O
rotation	O
applies	O
in	O
special	O
relativity	O
,	O
where	O
it	O
can	O
be	O
considered	O
to	O
operate	O
on	O
a	O
four-dimensional	O
space	O
,	O
spacetime	B-Protocol
,	O
spanned	O
by	O
three	O
space	O
dimensions	O
and	O
one	O
of	O
time	O
.	O
</s>
<s>
These	O
transformations	O
preserve	O
a	O
quadratic	O
form	O
called	O
the	O
spacetime	B-Protocol
interval	O
.	O
</s>
<s>
By	O
contrast	O
,	O
a	O
rotation	O
in	O
a	O
plane	O
spanned	O
by	O
a	O
space-like	O
dimension	O
and	O
a	O
time-like	O
dimension	O
is	O
a	O
hyperbolic	O
rotation	O
,	O
and	O
if	O
this	O
plane	O
contains	O
the	O
time	O
axis	O
of	O
the	O
reference	B-General_Concept
frame	I-General_Concept
,	O
is	O
called	O
a	O
"	O
Lorentz	O
boost	O
"	O
.	O
</s>
<s>
The	O
complex-valued	O
matrices	O
analogous	O
to	O
real	O
orthogonal	B-Algorithm
matrices	I-Algorithm
are	O
the	O
unitary	B-Algorithm
matrices	I-Algorithm
,	O
which	O
represent	O
rotations	O
in	O
complex	O
space	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
unitary	B-Algorithm
matrices	I-Algorithm
in	O
a	O
given	O
dimension	O
forms	O
a	O
unitary	O
group	O
of	O
degree	O
;	O
and	O
its	O
subgroup	O
representing	O
proper	B-Algorithm
rotations	I-Algorithm
(	O
those	O
that	O
preserve	O
the	O
orientation	O
of	O
space	O
)	O
is	O
the	O
special	O
unitary	O
group	O
of	O
degree	O
.	O
</s>
